Rational box-step functions form a countable dense subset of $L^p({\mathbb{R}}^n)$ for $1\le p<\mathrm{\infty }$

Let $\mathcal{R}$ be the family of half-open boxes [L2, L4, given, algebra] ${\prod }_{i=1}^n(a_i,b_i]$ with rational endpoints. By [L2] there are countably many such boxes, and by [L4] the set $\mathcal{S}$ of all finite rational linear combinations of indicators ${\mathbf{1}}_R$ with $R\in \mathcal{R}$ is countable.

By [L1], it is enough to approximate a single box indicator. So fix a [L1, L3, step 1.1, choose, algebra] $\delta >0$ and let $B={\prod }_{i=1}^n({\alpha }_i,{\beta }_i]$. Choose rationals $a_i<{\alpha }_i<{\beta }_i<b_i$ so close to the endpoints that, with $M:=1+{\max }_i\{\mathrm{\mid }{a}_i\mathrm{\mid },\mathrm{\mid }{b}_i\mathrm{\mid },\mathrm{\mid }{\alpha }_i\mathrm{\mid },\mathrm{\mid }{\beta }_i\mathrm{\mid }\}$, $2n(2M{)}^{n-1}{\max }_i\{({\alpha }_i-a_i),(b_i-{\beta }_i)\}<\delta \mathrm{.}$ Then $B\mathrm{△}R$ is contained in the union of the $2n$ coordinate slabs where one coordinate lies in $(a_i,{\alpha }_i]$ or $({\beta }_i,b_i]$ while the others stay in $[-M,M]$. By [L3], each slab has measure at most $(2M{)}^{n-1}{\max }_i\{({\alpha }_i-a_i),(b_i-{\beta }_i)\}$, so ${\lambda }_n(B\mathrm{△}R)<\delta$ for the rational box $R:={\prod }_i(a_i,b_i]$. Hence $\mathrm{\parallel }{\mathbf{1}}_B-{\mathbf{1}}_R{\mathrm{\parallel }}_p={\lambda }_n(B\mathrm{△}R{)}^{1\mathrm{/}p}<{\delta }^{1\mathrm{/}p},$ which can be made arbitrarily small; approximating finitely many coefficients by rationals then makes every box-step function arbitrarily close to an element of $\mathcal{S}$.

Therefore $\mathcal{S}$ is countable and dense. By [L5], [L5, step 1.1, step 2.1] $L^p({\mathbb{R}}^n)$ is separable, with $\mathcal{S}$ as an explicit dense subset.